A study of Z-magnitude dependence in the spatial orientation of angular momentum vectors of galaxies having redshift < 30,000 km s-1

I present a study of spin vector orientations of 229,675zmagnitude SDSS (Sloan Digital Sky Survey) galaxies having red shift of 0.05 to 0.10. The z-magnitudes are observed through 913.3 nm CCD (charge coupled device) filter attached to SDSS telescope located at New Mexico, USA. We have converted two dimensional data to three dimensional by Godlowskian Transformation. Our aim is to find out non-random effects in the spatial orientation of galaxies. The expected isotropy distribution curves are obtained by removing the selection effects and performing a random simulation method. In general, spin vector orientations of galaxies is found to be random, supporting Hierarchy model of galaxy formation. In some samples (z12 and z13) of polar angle distribution is observed anisotropy supporting Primordial Vorticity Theory. For azimuthal angle distribution in more sample we observed anisotropy result. A local anisotropy is observed in few samples suggesting a gravitational tidal interaction, merging process and gravitational lensing effect.

THE SPATIAL ORIENTATION OF ANGULAR MOMENTUM VECTORS OF GALAXIES HAVING

REDSHIFT < 30,000 km s-1

ABSTRACT

I present a study of spin vector orientations of 229,675z-magnitude SDSS (Sloan Digital Sky Survey) galaxies having red shift

of 0.05 to 0.10 The z-magnitudes are observed through 913.3 nm CCD (charge coupled device) filter attached to SDSS telescope located at New Mexico, USA We have converted two dimensional data to three dimensional by Godlowskian Transformation Our aim is to find out non-random effects in the spatial orientation of galaxies The expected isotropy distribution curves are obtained by removing the selection effects and performing a random simulation method In general, spin vector orientations of galaxies is found to be random, supporting Hierarchy model of galaxy formation In some samples (z12 and z13)

of polar angle distribution is observed anisotropy supporting Primordial Vorticity Theory For azimuthal angle distribution in more sample we observed anisotropy result A local anisotropy is observed

in few samples suggesting a gravitational tidal interaction, merging process and gravitational lensing effect

Key Words: clusters, Supercluster, galaxies formation, evolution of

galaxies, statistics, redshift, isotropy, anisotropy

INTRODUCTION

The initial perturbations in matter density which is believed to be present in early Universe, gradually kept on being enhanced with time as the Universe expanded and 1000s of millions of years after Big Bang slowly the large scale structures viz galaxies, clusters, Super clusters, Hyper clusters, filaments and walls, with great walls" of thousands of galaxies reaching more than a billion light years in length started getting formed However, less dense regions did not grow, evolving into an area

of seemingly empty space called voids Hereby, we see that our universe

is ever evolving and expanding; and cosmologists and astronomers are trying to discover what its final fate will be Since the universe by its definition encompasses all of the space and time as we know it, it is beyond the model of the Big Bang to say what the universe is expanding

* Mr Yadav is Reader in Physics at Central Department of Physics, Tribhuvan University, Kirtipur, Kathmandu, Nepal

into or what gave rise to the Big Bang Although there are models that speculate about these questions, none of them have made realistically testable predictions as of yet The Lambda Cold Dark Matter (or LCDM) model is the best model of our known universe about the origin of the large scale structures representing an improvement over the big bang

theory (Blumenthal et al., 1984) Large scale structure means the huge

cluster and Super cluster of galaxies These structures are one of the most mysterious discoveries, because their formation and evolution is not fully understandable yet Thus, to know the origin of the expanding universe and large scale structures is one of the most fundamental question in the recent universe One of the most accepted model on the evolution and expansion of the large scale structure is that it was the result of primordial fluctuations by gravitational instability An additional factor complicating

an understanding of galaxies is that their evolution is strongly affected by their environment The gravitational effects of other galaxies can be important Galaxies can sometimes interact and even merge

Modern cosmology is based on two fundamental assumptions: First, the dominant interaction on cosmological scales is gravity, and second, the cosmological principle is a good approximation to the universe The cosmological principle states that the universe, smoothed over large enough scales, is essentially homogeneous and isotropic

‘Homogeneity’ has the intuitive meaning that at a given time the universe looks the same everywhere and ‘isotropy’ refers to the fact that for any observer moving with the local matter, the universe looks (locally) the same in all directions Von Weizsacker and Gamow (1951 and 1952) made it clear that the observed rotation of the galaxies is important for cosmology: the fact that the galaxies rotation may be a clue to the physical conditions under which these systems are formed In the instability picture, one imagines that large irregularity like galaxies grew under the influence of gravity from small imperfections in the early Universe (Gamow and Teller 1939, Peebles 1965, 1967) In this picture one must abandon the idea that the angular momentum of the galaxies was given as the initial value, or developed in some sort of primeval turbulence(as was proposed by von Weizsaker1951 and Gamow 1952), for otherwise the galaxies would have formed too soon (Peebles, 1967) On the other hand, huge low red shift galaxy surveys such as the 2-degree field galaxy red

shift survey (2dFGRS, Colless et al 2001) and the Sloan Digital Sky

Survey (SDSS, York et al 2000) have convinced most cosmologists that not only isotropy but also homogeneity is, in fact, a reasonable assumption for the universe The Universe is homogeneous and isotropic on scales larger than 100 Mpc, but on smaller scales we observe huge deviations from the mean density in the form of galaxies, galaxy clusters, and the cosmic web being made of sheets and filaments of galaxies How do

structures grow in the universe and how we can describe them? The most accepted view on the formation and evolution of large scale structure is that it was formed as a consequence of the growth of primordial fluctuations by gravitational instability In the current favored model, smaller structures collapse first and are later incorporated in larger collapsing structures in a bottom-up scenario that provides a natural explanation for the formation of galaxies, clusters, filaments and Super clusters

Galaxy clusters are gravitationally bound large scale structures in the universe To understand the evolution of these aggregates, it is essential to know when and how they were formed and how their structures and constituents have been changing with time Gamow (1952) made it clear that the observed rotations of galaxies are important for cosmology According to them, the fact that galaxies rotation might be a clue of physical conditions under which these systems are formed Thus, understanding the distribution

of spatial orientations of the spin vectors (hereafter SVs) or angular momentum vectors of galaxies is very important It could allow us to know the origin of angular moment of a galaxy

There are three predictions about the spatial orientation of spin

vectors of galaxies These are the `pancake model', the `hierarchy model,' and the `primordial vorticity theory.' The `pancake model'

(Doroshkevichmm, 1973; Doroshkevich and Shandarin, 1978) predicts that the spin vectors (SVs hereafter) of galaxies tend to lie within the cluster plane According to the `hierarchy model' (Peeblesm, 1969), the directions of the SVs should be distributed randomly The `Primordial Vorticity Theory' (Ozernoy, 1971, 1978; Stein, 1974) predicts that the spin vectors of galaxies are distributed primarily perpendicular to the cluster plane

SDSS PHOTOMETRY

The term “photo” means light and the term “metry” refers to measurements Photometry is the science of the measurement of light, in term of its perceived brightness to the human eye In photometry, the standard is the human eye The sensitivity of the human eye to different color is different This has to consider in photometry Since the human eye

is only sensitive to visible light, photometry only falls in that range Photometry and spectroscopy are two important applications of light measurements These two methods have various applications in fields such as chemistry, physics, optics and astrophysics It is vital to have a solid understanding in these concepts in order to excel in such fields

The apparent magnitudes do not tell us about the true brightness

of stars, since the distances differ A quantity measuring the intrinsic brightness of a star is the absolute magnitude The absolute magnitude M

of any star is defined as the apparent magnitude at a distance of 10 pc from the star in the absence of astronomical extinction

We know that the relation between magnitude, distance and

opacity is given by Padmanabhan (2006),

Here m, M and α represent apparent, absolute magnitudes and

opacity of the medium

For r and u filter,

Subtracting equation (3) from (2) we get,

r = [(m r - m u )-(M r - M u )]/[2.5 loge (α r -α u )] (4)

This equation is useful to calculate the value of distance to the galaxy Thus, the magnitude through various filters gives precise information about distance The SDSS Telescope consists of five different filters of different pass band and different wave lengths given in Figure-1:

Figure-1: Wave length bands of the filters used by SDSS telescope

GODLOWSKIAN TRANSFORMATION

The three dimensional orientation of the SV of a galaxy is characterized by two angles: the polar angle (θ between the galactic SV

and a reference plane (here equatorial plane), and the azimuthal angle φ between the projection of a galactic SV on to this reference plane and the X-axis within this plane The detail derivations of the expressions of the

angles θ and φ are given in Flin and Godłowski (1986) When using equatorial coordinate system as reference, then θ and φ can be obtained

from measurable quantities as follows:







Where, i is the inclination angle, the angle between the normal to

the galaxy plane and the observer’s line-of-sight, α- is right ascension, δ is declination and p is position angle The inclination angle can be computed from the formula,

)

*

1

(

)

* (

2 2

2

q

q

q

i





This expression is valid for oblate spheroids (Holmberg 1946) Here,

q and q* represent the measured axial ratio (b/a) and the intrinsic flatness of the galaxy, respectively Hiedmann et al (1971) showed that the values of q*

range from 0.083 for Sd spirals to 0.33 for ellipticals For the galaxies with

unknown morphology q*=0.20 is assumed Holmberge (1946)

NUMERICAL SIMULATION

Here we describe the procedure for the removal the selection effects to obtain the isotropic distributions for both θ and φ as given by Aryal and Saurer (2000) Theoretically, the isotropic distribution curve for polar angle is cosine and that for azimuthalangle is the average distribution curve, with the restriction that the database is free from selection effect Aryal and Saurer (2000) concluded that any selections imposed on the database may cause severe changes in the shapes of the expected isotropic distribution curves In their method, a true spatial distribution of the galaxy rotation axis is assumed to be isotropic Then,

due to the projection effects, i can be distributed as ~sin i, B can be

distributed ~cos B, the variables α and p can be distributed randomly, and

the equation (5,6) can be used to calculate the corresponding values of

polar (θ) and azimuthal (φ) We run simulations in order to define expected isotropic distribution curves for both the θ and φ distributions

The isotropic distribution curves are based on simulations including 106 virtual galaxies At first we observed the distributions of α, δ, p and i for the galaxies in our samples and distributed by creating 106 virtual galaxies for respective parameters We use these numbers to make input file and the expected distribution by running simulation in MATLAB 7.0

METHOD OF ANALYSIS

Our observations (real observed data set) are compared with the

isotropic distribution curves (obtained from simulation) in both θ and φ

distributions For this comparison we use three different statistical tests: chi-square-, Fourier-, and auto correlation-test

C HI - SQUARE TEST

The Chi squire test is a measure of how well a theoretical

probability distribution fits a set of data Chi squire test is an objective way to examine whether the observational distribution deviates from the expected distribution (in our case, to the isotropic distribution), which is based on the reduced value given by,











 n

ei i

N

N N

1

2 0

Here, n represents the number of bins, N 0i and N ei represent the observed and expected isotropic distributions and ν is the degree of

freedom (ν = n-1) For an isotropic distribution, the 2



 value is expected

to be nearly zero The quantity P(> 2



 ) gives the probability that the observed 2



 value is realised by the isotropic distribution The observed distribution is more consistent with the expected isotropic distribution when P(> 2



 ) is larger We set P(> 2



 ) =0.050 as the critical value to discriminate isotropy from anisotropy, it corresponds to the deviation from the isotropy at 2≤ σ level

F OURIER TEST

We compare the observed distributions of the polar and azimuthal

angle (θ and φ) of the galaxy rotation axis with the expected isotropic

distributions If the deviation from isotropy is a slowly varying function of

the angle θ (or φ), one can use the Fourier test

Here, the1st order Fourier coefficients ∆11 is given by













k

k k

n

k

k k

k

N

N N

1

2 0

1

0 11

2 cos

2 cos ) (





(10)

The standard deviations σ (∆11) can be obtained using the expressions

2 / 1 2

1 0

(



















n

k k

The Fourier coefficient ∆11 is very important because it gives the

direction of departure from isotropy In the analysis of the polar angle (θ),

if ∆11<0 an excess of galaxies with rotation axis parallel to the reference plane is present, whereas for ∆11>0 the rotation axis tend to be

perpendicular to that plane In the analysis of the azimuthal angle (φ), if

∆11<0 an excess of galaxies with rotation axis projections perpendicular to the direction of the Virgo cluster center, whereas with ∆11>0 the projections tend to be parallel to that direction

A UTO CORRELATION TEST

Auto correlation test is a measure of a degree to which there is a linear relationship between two variables In our case, it takes account of the correlation between the numbers of galaxies in adjoining angular bins The correlation function is







 



k k

k k

k k

N N

N N N N C

1 0 0 1 1/2

1 0 1 0

)

with the standard deviation  1 / 2

) (C  n



In an isotropic distribution any correlation vanishes, so we expect

to have C0

R ELIABILITY OF S TATISTICS IN G ALAXY O RIENTATION S TUDY

It is clear from the expression (9) that the χ 2 value increases when

the number of solutions per bin for both the observed (N 0i) and the

expected (N ei ) distributions is increased by the same factor As the χ 2value

increases, P (> χ 2) decreases So, our result 'isotropic' can change into 'anisotropic', if the database is filled with additional galaxies However, the result 'anisotropic' remains the same even if the database increases Because of the incompleteness of database the numbers we deal with are lower limits So, our result 'anisotropic' can be seen as more reliable than the result 'isotropic' The same will happen in the case of the correlation

coefficient C and the Fourier probability function P (>∆1) However, the Fourier coefficient ∆11/σ(∆11) value remains unchanged even if the database increases

RESULTS AND DISCUSSION

Any deviation from expected isotropic distribution will be tested using four statistical parameters, namely chi-square probability (P >χ2), autocorrelation coefficient (C/C(σ)), first order Fourier coefficient (∆11/σ(∆11)) and first order Fourier probability (P >∆1) For anisotropy, the limit of chi-square probability P(>χ2) is <0.050, auto correlation coefficient (C/C(σ)) is >1.0, first order Fourier coefficient (∆11/σ(∆11)) is

>1.5 and Fourier probability P(>∆1) is <0.150, respectively Any `hump' (more solutions than the expected) or `dip' (less solutions than the

expected) in the histogram will be discussed as a local effect in the samples The statistics for the polar and azimuthal angle distributions is

given in Table-1 In the statistics of θ, a negative value of first order

Fourier coefficient suggests that the spin vectors of galaxies tend to be oriented perpendicular with respect to the equatorial coordinate system Similarly, a positive value of first order Fourier coefficient suggests that the spin vectors of galaxies tend to be oriented parallel with respect to the

equatorial coordinate system Whereas, in the statistics of φ, a positive

(∆11/σ(∆11)with significant value suggests that the spin vector projections

of galaxies tend to point radically with respect to the center of the equatorial coordinate system Similarly, a significant negative value of (∆11/σ(∆11)implies that the spin vector projection of galaxies tend to orient tangentially with respect to the equatorial coordinate system

In addition to the statistical tests, we also study the `humps' and

`dips' in the polar and azimuthal angle distributions The solid curve, in

the histogram of the φ-distribution, represents the expected isotropic

distribution whereas dashed curve is the cosine distribution The solid circles with ±1σerror bars represent the observed distribution The shaded portion represents the range 0o<θ<45o A hump (or dip) in the smaller suggests that the spin vectors of galaxies tend to orient parallel (or perpendicular) with respect to the equatorial coordinate system

Similarly, a hump (or dip) in the larger θ indicates that the spin

vectors of galaxies tend to be oriented perpendicular with respect to the

equatorial coordinate system In the histogram of the θ-distribution, solid

curve represents the expected isotropic distribution whereas dashed curve

is the average distribution The solid circles with ±1σ error bars represent the observed distribution The shaded portion represents the range -45o< φ

<+45o The humps and dips in the histograms of φ distribution are not so easy to interpret as compared to θ-distributions It is because the range of

φ is -90oto +90o In the histogram of the -distribution, φ = 0omeans spin vector projections tend to point radially towards the center of the equatorial coordinate system A hump in the middle (central eight bins) of the histogram suggests that the spin vector projections of galaxies tend to point towards the center of the chosen co-ordinate system Similarly, a hump at first four and last four bins indicates that the spin vectors projections of galaxies tend to be oriented tangentially with respect to the chosen reference co-ordinate system

This sample contains a small number (51) of galaxies that have apparent magnitude in the range 13.25 to 13.55 in the z-band (913.4 nm wavelength) The smaller the value of magnitude the brighter the object Significant infrared emission indicates the enhanced star formation activity and the relatively higher abundance of metal-rich elements in these galaxies Thus, these are the brightest galaxies in the z-filter Figure,

2 shows the polar angle (θ) distribution of total galaxies of database in the sample z01 The statistics for the θ-distribution of galaxies of this sample

is shown in the Table-1 The statistics for the polar angle distribution in this sample shows that the value of chi-square probability (P(>χ2)) to be 0.458 i.e., 45.8% (greater than the significant level 0.050 i.e., 5.0%) The auto-correlation coefficient (C/C(σ)) is found to be 1.4 (greater than 1σ limit) The first order Fourier coefficient (Δ11/σ(Δ11))is found to be +0.5 (smaller than 1.5σ the limit) The first order Fourier probability (P>(1)) is found to be 0.849 i.e., 84.9% (greater than 15% limit) Except auto-correlation coefficient, other statistical tests suggest isotropy Anisotropy

in the auto-correlation test suggests either binning effect or local anisotropy that should be seen in the humps/dips in the histogram In, Figure-2, the number of observed solutions that have θ< 450 is found to be equal to the expected solutions i.e., 64 There is no any significant humps and dips in the lower angles (<450) region At bimodal region (θ≈450), there are 2 more observed solutions than the expected in this region For the large angles (> 450), the number of expected solutions is more by 2 than that of observed and there is a significant dip at 63.750 with >2σ error limit in this region Thus, we conclude that there is no preferred alignment

of spin vectors of galaxies that are brightest when observed through filter

The statistics of the φ-distribution (Table-1) for the sample z01 brings the results as: P (>χ2) = 0.187 i.e., 18.7% (greater than the 5.0% significant level), C/C(σ) = 0.0 (less than the limit 1σ), Δ11/σ(Δ11) = 1.6 (greater than the limit 1.5σ), P >(Δ1) = 0.223 i.e.,22.3% (greater than 15% limit) The values of chi-square probability (P(>χ2) ), the auto-correlation coefficient C/C(σ) and the first order Fourier probability support strong isotropy However first order Fourier coefficient suggests the weak anisotropy In the azimuthal angle distribution, as shown in Figure-2, the observed solutions of central six bins (shaded region) are found to be 65, whereas the expected solutions are only 57 This shows that observed solutions exceeded the expected solutions by 8 In this region, one dip at

an angle 7.50is observed with ≈1.5σ error limit and two humps are observed at 22.50 and 7.50with≈1.5σ error limit Since, there are no significant dips and humps outside the shaded part Owing to the extremely poor statistics (i.e., the number of solutions per bin), we conclude no preferred alignments The bright infrared galaxies that have red shift in the range 0.05 to 0.10 showed no preferred alignments in the spatial orientation of their spin vectors

Table-1: Statistics of the polar (left) and azimuthal angle (right)

distributions of z-magnitude SDSS galaxies having red shift in

the range 0.01 to 0.10in the samples The P(>χ2) represents the chi-square probability (second column) Similarly, C/C(σ) represents the auto-correlation coefficient (third column) The last two columns give the first order Fourier coefficient (∆11/σ(∆11) and first order Fourier probability P(>∆1)